One of the biggest winners in the sweeping Democratic victories in Senate races this month was a woman whose name appeared on no ballot: Sen. Patty Murray (D-Wash.).

With a low-key style that contrasts with some of the Senate’s camera hogs, Murray may be the most powerful senator a whole lot of people have never heard of outside of the two Washingtons where she lives and works.

As chair of her party’s Senate campaign arm, the architect of surprising Democratic gains and the incoming chair of the powerful Senate Budget Committee, Murray now occupies a place of special influence in the Senate.

And so what Murray has to say about the “fiscal cliff,” a combination of tax increases and spending cuts set to take effect in January, may be of particular importance. In a town consumed by talk of the apocalyptic consequences of failing to resolve the budgeting crisis, Murray has been arguing that missing the deadline for a deal — going over the cliff — could actually make getting a deal easier.

“She’s low-key but very focused and very forceful,” said Sen. Claire McCaskill (D-Mo.), whose defeat was widely predicted but who won reelection in November after her Republican opponent claimed that pregnancy rarely results from rape. “If you know Patty and you work with her, you’d be a fool to underestimate her.”

Two years ago, she agreed to chair the Democrats’ 2012 senatorial campaign effort. At the time, it was widely assumed the job would mean presiding over the party’s loss of its majority. Republicans needed only four seats to prevail, and Democrats were defending 23 seats, compared with 10 for the GOP. Key retirements in several Republican-leaning states seemed to portend a GOP takeover.

Instead, through a combination of good candidate recruitment, good luck and Republican missteps, Democrats won almost every tough contest, expanded their effective majority by two seats and helped achieve several historic firsts, including a record number of women elected to the Senate.

In a chamber of big egos, Murray’s success at achieving what had been thought impossible has given her new leverage with her Senate colleagues, who are especially grateful that no Democratic incumbent lost.

Now, Murray is using that influence to argue that Democrats should not forget the tactical advantage they could gain in January, after the deadline for the fiscal cliff has passed.

Starting with a speech at the Brookings Institution in July and continuing in a series of interviews last week, Murray, in her typically non-bombastic fashion, has argued that Democrats shouldn’t take a bad deal in December when their political leverage will only increase in the new year.

That’s because next month, tax cuts first enacted under President George W. Bush will expire for everybody. Murray reasons that might make it easier to get Republicans to agree to reinstate the cuts only for the middle class and let the nation’s wealthiest 2 percent pay more toward the reducing the debt, as Democrats desire.

Suddenly, Congress would be voting to lower taxes for the vast majority of Americans, rather than agreeing to raise them for a few.

“I’m hopeful we can get there. I’m an optimist” about a deal to avoid the cliff, she said in an interview. But, she added, “the dynamic changes dramatically on January 1st. . . . Anything we do will be a tax cut at that point, because taxes will have gone up.”

In a Congress of hot tempers and sharp tongues, Murray doesn’t favor over-the-top rhetoric. Once dismissed by a Washington state representative as just a “mom in tennis shoes,” she’s turned the moniker into a campaign symbol of determined strength.

Murray, colleagues agree, doesn’t issue idle threats.

“Everyone takes Senator Murray seriously because she does not bluster,” Reid said. “She simply says what she means and stands by it.”

Her perspective is born in part from the last tough task Reid handed her, chairing last year’s bipartisan “supercommittee,” a 12-member panel that tried, but ultimately failed, to come up with a major deficit reduction package acceptable to both parties.

Murray spent long hours behind closed doors with House and Senate Republicans and emerged convinced the GOP was offering only damaging proposals to cut health, education and environmental programs without agreeing to ask the wealthy to pay more in taxes.

“The experiences she has had and the leadership she has shown carry a lot of weight with her fellow senators. When you see someone of her reasonableness and moderation saying something, you take it seriously,” Reid said.

The episode gave Murray new cachet with liberal allies who are now nervous the White House could give too much to get a deal to avoid going over the cliff, agreeing to major cuts to Medicare or Social Security.

If President Obama does reach a deal with House Speaker John A. Boehner (R-Ohio) this month, Murray’s blessing could boost a package that might otherwise be hard for fellow senators to swallow.

Likewise, her objections could stop a developing deal in its tracks.

Murray has quietly played that role twice before in the monumental budget debates that have unfolded since Republicans swept into control of the House in 2010.

In the crunch of final negotiations over a deal to raise the nation’s debt ceiling last summer, it was Murray who nixed the idea of exposing veterans benefits to automatic domestic and military spending cuts that would result if Congress does not reach a more targeted deficit-reduction deal by the end of this month.

“Joe, Patty Murray is one of the driving forces in my caucus. If she doesn’t like it, she’ll kill the bill,” Reid told Vice President Biden when he called to broach the idea during talks. He then turned to Murray, who was seated across from him. “Patty, do you like it?”

It was Murray, too, who had counseled Democrats in April 2011 to reject a last-minute demand by House Republicans to eliminate federal funding for Planned Parenthood in a spending bill designed to avoid a government shutdown.

Murray recalled she had gone home for the evening convinced a deal was in sight. She received a call from Reid asking her to return to a late-night meeting at the Capitol to discuss the Republicans’ final request.

“I walked in, and I was literally the only woman,” she remembered. “And I walked in and they said: ‘We’re all done except the House wants one last concession. They want us to give on that and we’re done.’

“And I said: ‘Not on my watch. Absolutely not on my watch,’ ” she said.

Murray rallied female senators to take the floor early the next morning to blast the idea. Again, her advice to hold firm at a key moment bore fruit: Democrats ultimately forced the GOP to give on the issue.

Behind the scenes, Murray was credited by colleagues with recognizing that female voters in key races could be galvanized around the idea that Republicans were more concerned about limiting access to abortion and contraception than promoting jobs and the economy.

It proved to be an exceptionally powerful campaign message, especially after Republicans in Missouri and Indiana made intemperate remarks about rape and abortion.

“She strategically saw how, not only was it the right thing to do, but that it was politically important for us to make that case to people,” said Guy Cecil, executive director of the Democratic Senatorial Campaign Committee.

Now, she’s preparing to take on another job thought thankless by many — chairman of the Senate Budget Committee. She will succeed retiring Sen. Kent Conrad (D-N.D.), a highly respected centrist who nevertheless drew fire from Republicans for presiding over a committee that did not get a budget off the Senate floor in three years.

Sen. Jeff Sessions (Ala.), the committee’s ranking Republican, praised Murray for a reputation for moderation and common sense. But, he warned, the committee’s functioning will not improve if Murray is beholden to Reid.

“I think one of the challenges Senator Murray will have is will she move forward with a budget? Or will the majority leader prefer not to have a budget and will she go along with that?” Sessions said.

Murray has not guaranteed the committee will produce a budget under her leadership, although she said she hopes debt talks underway could result in a spending agreement between the parties that will last into next year.

“I think the budget challenge is the same as everything I’ve ever done. You can say, ‘Gosh, that’s hard,’ and not agree to take it on,” she said.

“Then you don’t have the ability to impact it in a way that’s important to you.”